Dynamics Models for Tuberculosis Transmission and Control

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Dynamics Models for Tuberculosis Transmission and
Control
Carlos Castillo-Chavez
Department of Biological Statistics and
Computational Biology Department of Theoretical
and Applied Mechanics Cornell University,
Ithaca, New York, 14853
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Ancient disease

• TB has a history as long as the human race.
• TB appears in the history of nearly every
  culture.
• TB was probably transferred from animals to
  humans.
• TB thrives in dense populations.
• It was the most important cause of death up to
  the
• middle of the 19th century.

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Transmission Process

• Causative agent
• Tuberculosis Bacilli (Koch, 1882).
• Preferred habitat
• Lung.
• Main Mode of transmission
• Host-air-host.
• Immune Response
• Immune system tends to respond quickly to
  initial invasion.

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Immune System Response Caricature

• Bacteria invades lung tissue.
• White cells surround the invaders and try to
  destroy them.
• Body builds a wall of cells and fibers around the
  bacteria to confine them, forming a small hard
  lump.

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Immune System Response Caricature

• Bacteria cannot cause additional damage as long
  as confining walls remain unbroken.
• Most infected individuals never develop active TB
  (that is, become infectious).
• Most remain latently-infected for life.
• Infection progresses and develops into active TB
  in less than 10 of the cases.

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TB was the main cause of mortality

• Leading cause of death in the past.
• Accounted for one third of all deaths in the 19th
  century.
• One billion people died of TB during the 19th and
  early 20th centuries.
• TBs nicknames White Death, Captain of Death,
  Time bomb

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Per Capita Death Rate of TB
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Current Situation

• Two to three million people around the world die
  of TB each year.
• Someone is infected with TB every second.
• One third of the world population is infected
  with TB ( the prevalence in the US is 10-15 ).
• Twenty three countries in South East Asia and Sub
  Saharan Africa account for 80 total cases around
  the world.
• 70 untreated actively infected individuals die.

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TB in the US
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Reasons for TB Persistence

• Co-infection with HIV/AIDS (10 who are HIV
  positive are also TB infected).
• Multi-drug resistance is mostly due to incomplete
  treatment.
• Immigration accounts for 40 or more of all new
  recent cases.
• Lack of public knowledge about modes of TB
  transmission and prevention.

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Earliest Models

• H.T. Waaler, 1962
• C.S. ReVelle, 1967
• S. Brogger, 1967
• S.H. Ferebee, 1967

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Epidemiological Classes
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Parameters
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Basic Model Framework

• NSEIT, Total population
• F(N) Birth and immigration rate
• B(N,S,I) Transmission rate (incidence)
• B(N,S,I) Transmission rate (incidence)

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Model Equations
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Epidemiology(Basic Reproductive Number, R0)

• The expected number of secondary infections
• produced by a typical infectious individual
• during his/her entire infectious period
• when introduced in a population of mostly
• susceptibles at a demographic steady state.
• Sir Ronald Ross (1911)
• Kermack and McKendrick (1927)

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Epidemiology(Basic Reproductive Number, R0)

• Frost (1937) wrote it is not necessary that
  transmission be immediately and completely
  prevented. It is necessary only that the rate of
  transmission be held permanently below the level
  at which a given number of infection spreading
  (i.e. open) cases succeed in establishing an
  equivalent number to carry on the succession

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R0

• Probability of surviving the latent stage
• Average effective contact rate
• Average effective infectious period

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Demography
F(N)?, Linear Growth
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Exponential Growth(Three Thresholds)

• The Basic Reproductive Number is

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Demography and Epidemiology
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Demography
Where
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  Bifurcation Diagram (exponential growth )
 
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Logistic Growth
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Logistic Growth (contd)

• If R2 gt1
• When R0 ? 1, the disease dies out at an
  exponential rate. The decay rate is of the order
  of R0 1.
• Model is equivalent to a monotone system. A
  general version of the Poincaré-Bendixson Theorem
  is used to show that the endemic state (positive
  equilibrium) is globally stable whenever R0 gt1.
• When R0 ? 1, there is no qualitative difference
  between logistic and exponential growth.

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Bifurcation Diagram
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Particular Dynamics(R0 gt1 and R2 lt1)
All trajectories approach the origin. Global
attraction is verified numerically by randomly
choosing 5000 sets of initial conditions.
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Fast and Slow TB (S. Blower, et al., 1995)
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Fast and Slow TB
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Variable Latency Period (Z. Feng, et al,2001)
p(s) proportion of infected (noninfectious)
individuals who became infective s unit of time
ago and who are still infected (non infectious).
Number of exposed from 0 to t who are alive and
still in the E class
Number of those who progress to infectious from
0 to t and who are still alive in I class at time
t
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Variable Latency Period (differentio-integral
model)

• E0(t) of individuals in E class at t0 and
  still in E class at time t
• I0 of individuals in I class at t0
  and still in I class at time t

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Exogenous Reinfection
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Exogenous Reinfection
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Backward Bifurcation
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Age Structure Model
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Parameters

• ? recruitment rate.
• ?(a) age-specific probability of becoming
  infected.
• c(a) age-specific per-capita contact rate.
• ?(a) age-specific per-capita mortality rate.
• k progression rate from infected to
  infectious.
• r treatment rate.
• ? reduction proportion due to prior exposure
  to TB.
• ? reduction proportion due to vaccination.

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Proportionate Mixing

• p(t,a,a) probability that an individual of
  age a has
• contact with an individual of age a given
  that it has
• a contact with a member of the population .
• Proportionate mixing p(t,a,a) p(t,a)

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Incidence and Mixing
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Basic reproductive Number (by next generation
operator)
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Stability
There exists an endemic steady state whenever
R0(?)gt1. The infection-free steady state is
globally asymptotically stable when R0 R0(0)lt1.
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Optimal Vaccination Strategies

• Two optimization problems
• If the goal is to bring R0(?) to pre-assigned
  value
• then find the vaccination strategy ?(a) that
  minimizes the
• total cost associated with this goal (reduced
  prevalence to a
• target level).
• If the budget is fixed (cost) find a vaccination
  strategy ?(a)
• that minimizes R0(?), that is, that minimizes the
  prevalence.

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Optimal Strategies

• Oneage strategy vaccinate the susceptible
  population
• at exactly age A.
• Twoage strategy vaccinate part of the
  susceptible
• population at exactly age A1 and the
  remaining
• susceptibles at a later age A2.
• Optimal strategy depends on data.

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Challenging Questions associated with TB
Transmission and Control

• Impact of immigration.
• Antibiotic Resistance.
• Role of public transportation.
• Globalizationsmall world dynamics.
• Time-dependent models.
• Estimation of parameters and distributions.